3.1: Constants and Variables

Learning Outcomes

Use variables to represent unknown quantities in algebraic expressions
Identify the variables and constants in an algebraic expression
Use symbols and words to represent algebraic operations on variables and constants

Key words

Constant: a number that never changes
Variable: a letter used as a stand-in for a number that can change

Using Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is $20$ years old and Alex is $23$ , so Alex is $3$ years older than Greg. When Greg was $12$ , Alex was $15$ . When Greg is $35$ , Alex will be $38$ . No matter what Greg’s age is, Alex’s age will always be $3$ years more, right?

In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The $3$ years between them always stays the same, so the age difference is the constant.

In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age $g$ . Then we could use $g+3$ to represent Alex’s age. See the table below.

Greg’s age	Alex’s age
$12$	$15$
$20$	$23$
$35$	$38$
$g$	$g+3$

Variables and Constants

A variable is a letter that represents a number or quantity whose value may change.
A constant is a number whose value always stays the same.

To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. Earlier, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.

Operation	Notation	Say:	The result is…
Addition	$a+b$	$a\text{ plus }b$	the sum of $a$ and $b$
Subtraction	$a-b$	$a\text{ minus }b$	the difference of $a$ and $b$
Multiplication	$a\cdot b,\left(a\right)\left(b\right),\left(a\right)b,a\left(b\right)$	$a\text{ times }b$	The product of $a$ and $b$
Division	$a\div b,a/b,\frac{a}{b},b\overline{)a}$	$a$ divided by $b$	The quotient of $a$ and $b$

In algebra, the cross symbol, $\times$ , is no longer used to show multiplication because that symbol may cause confusion with the letter x, or variable $x$ . Does 3xy mean $3\times y$ (three times $y$ ) or $3\cdot x\cdot y$ (three times $x\text{ times }y$ )? To make it clear to everyone, we use a center dot • or parentheses ( ) to represent multiplication in algebra.

Translating from words to algebra or vice versa is an important skill. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help find the numbers.

The sum of $5$ and $3$ means addition, $5$ plus $3$ , which we write as $5+3$ .
The difference of $9$ and $2$ means subtraction, $9$ minus $2$ , which we write as $9 - 2$ .
The product of $4$ and $8$ means multiplication, $4$ times $8$ , which we can write as $4 \times 8$ .
The quotient of $20$ and $5$ means division, divide $20$ by $5$ , which we can write as $20\div 5$ .

Exercises

Translate from algebra to words:

$12+14$
$\left(30\right)\left(5\right)$
$64\div 8$
$x-y$

Solution:

12 + 14

$12+14$

12

$12$ plus

14

$14$

the sum of twelve and fourteen

(30) (5)

$\left(30\right)\left(5\right)$

30

$30$ times

5

$5$

the product of thirty and five

64 \div 8

$64\div 8$

64

$64$ divided by

8

$8$

the quotient of sixty-four and eight

x - y

$x-y$

x

$x$ minus

y

$y$

the difference of

x

$x$ and

y

$y$

TRY IT

When two quantities have the same value, we say they are equal and connect them with an equals sign.

Chapter 3: Variables, Expressions and Equations